Question #40355

Let $t \in \mathbb{R} \setminus \{0\}$ be a parameter. Define the determinants: $D_1(t) = \begin{vmatrix} t^2 & 3t & 5 \\ 1 & 1 & 0 \\ 0 & 1 & 1 \end{vmatrix} \quad \text{and} \quad D_2(t) = \begin{vmatrix} t^2 & 3t & -3 \\ 1 & 1 & 0 \\ 0 & 1 & 1 \end{vmatrix}$ Consider the rational functions $f_1(x) = \frac{t x^2 - D_1(t) x + 1}{x + 1}$ and $f_2(x) = \frac{x^2 + D_2(t) x + 1}{t x - t^2}$. Let $L_1$ and $L_2$ be the oblique asymptotes of $f_1(x)$ and $f_2(x)$ respectively. Let $S$ be the set of all integers $t$ such that $L_1$ and $L_2$ intersect at a unique point $P(x_t, y_t)$, and both $x_t$ and $y_t$ are integers. Find the sum of all elements in $S$.